CN107085865B - Quadrilateral segmentation method applied to finite element analysis - Google Patents

Quadrilateral segmentation method applied to finite element analysis Download PDF

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CN107085865B
CN107085865B CN201710335308.5A CN201710335308A CN107085865B CN 107085865 B CN107085865 B CN 107085865B CN 201710335308 A CN201710335308 A CN 201710335308A CN 107085865 B CN107085865 B CN 107085865B
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徐岗
舒来新
许金兰
朱亚光
潘德燃
吴卿
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Hangzhou Dianzi University
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Abstract

The invention discloses a quadrilateral segmentation method applied to finite element analysis. The division of the quadrilateral mesh is not implemented in various mainstream software. Firstly, changing a multi-connected polygon into a single connection, then reducing the number of fixed points of a boundary and extracting geometric characteristic points of the boundary in the process of simplifying the boundary; decomposing the original polygon into a plurality of polygons which have relatively simple geometric shapes and are convenient for topology generation and mapping; the topology generation adopts a fixed topology mode to match different sub-polygons to complete the construction of a topology structure; and finally, obtaining a final grid by using an overrun interpolation mapping method. The invention provides a high-quality quadrilateral mesh subdivision method. The quadrilateral mesh generated by the method has fewer singular points, has good smoothness, uniformity and orthogonality, and can be directly applied to finite element analysis simulation solution.

Description

Quadrilateral segmentation method applied to finite element analysis
Technical Field
The invention belongs to the field of computer aided design and engineering application, and particularly relates to a quadrilateral segmentation method of a planar polygon applied to finite element analysis.
Background
Quadrilateral mesh is a common mesh in the problem of two-dimensional space scientific computation, and the research on the quadrilateral mesh subdivision method in a two-dimensional plane domain is also a research subject which is very concerned at present. Meanwhile, many scholars have conducted a great deal of research on the quadrilateral mesh subdivision method, and many subdivision algorithms or improved algorithms are proposed.
At present, the division of the quadrilateral mesh is divided from the implementation approach, and can be divided into two types, one type is indirect method, and the other type is direct method. The indirect method is to generate a triangular mesh first and then convert the triangular mesh into a quadrilateral mesh by a certain algorithm. The direct rule is to directly divide the two-dimensional geometric definition domain into quadrilateral meshes by methods such as mapping or area decomposition. The mapping method is a relatively mature grid division method, and has also received high attention from both academic and industrial fields. Although the mapping method has the defects, the algorithm is simple, the unit quality is good, the density is controllable, and the mapping method can generate both structured grids and unstructured grids. Therefore, the mapping method still plays an important role in many quadrilateral mesh partitioning methods.
However, the above methods are all theoretical implementations, and are not implemented and applied in various mainstream software.
Disclosure of Invention
The invention aims to provide a quadrilateral meshing method applied to finite element analysis for a given polygon boundary in a two-dimensional plane domain aiming at the defects of the prior art, and the meshing process is completed mainly by means of the idea of a mapping method.
The method comprises the following specific steps:
step 1, splitting a geometric body with a through hole, wherein the cross section obtained after the geometric body is split is a plane figure which is provided with only one inner boundary and one outer boundary, and the inner boundary and the outer boundary are both polygons or are formed by adopting polygon fitting.
Step 2, converting the multi-connected domain into a single-connected domain;
the set of all the vertexes on the inner boundary of the obtained plane graph is a point set V1The set of all the top points on the outer boundary is the point set V2. Set of points V1And V2Find point v thereina、ubL (v) of the othera,ub) Is the smallest of all inner-boundary vertex-to-outer-boundary vertex distances, i.e.:
l(va,ub)＝min(l(vi,uj)),vi∈V1,uj∈V2
wherein i is more than or equal to 0 and less than or equal to n1-1, j is more than or equal to 0 and less than or equal to n2-1, n1 is an integer more than or equal to 3, and n2 is an integer more than or equal to n 1.
In obtaining va,ubThen, from the set of points:
V1＝{v0,v1,...,vn1-1}
V2＝{u0,u1,...,un2-1}
the sequences of the point set V' are obtained:
V′＝{v0,v1,...,va,ub,ub+1,...,un2-1,u0,u1,...,ub,va,va+1,...,vn1-1}
then the edge (v)a,ub) And (u)b,va) Is a pair of newly added edges with overlapped positions, and the number of the points in the point set V' is as follows: n is n1+n2+2. Let the set of all edges of the plane graph be the edge set E. Let simply connect the boundary M ═ V', E.
Opposite side (v)a,ub) And side (u)b,va) Performing uniform interpolation, i.e. at the edge (v)a,ub) And side (u)b,va) A new set of vertices Z is added. l (v)a,ub) The value m divided by the average length of all edges in the edge set E is calculated as follows:
Figure BDA0001293647390000021
where p is (i +1) mod (n1), q is (i +1) mod (n2), mod represents a remainder calculation, and the edge set E is a set of edges1Is V1Set of lengths of the lines between adjacent points in (E)2Is V2The length of the line segment between each adjacent point in the set, the number m of the top points of the top point set Z1Rounding off the rounding value for m.
Sequence of point set V:
Figure BDA0001293647390000022
in the formula (I), the compound is shown in the specification,
Figure BDA0001293647390000023
are vertices in the set of vertices Z.
And 3, simplifying the plane graph, which comprises the following specific steps:
a. deleting may simplify vertices:
① order simplified point set Vs＝V。
② two thresholds for controlling the degree of simplification of the boundary are defined as a simplified angle A and a simplified area ratio gamma, wherein A is more than or equal to 0 and less than or equal to pi, and gamma is more than or equal to 0 and less than or equal to 0.25.
③ set of points V1At any point viAnd two adjacent vertexes with viIs the apex angle αiThe included angle of the vertex of the outer boundary is formed, and a triangle formed by the three points is the triangle of the vertex of the outer boundary; set of points V2At any point ujAnd two adjacent vertices by point ujIs the apex angle βjThe included angle of the vertex of the inner boundary is shown, and the triangle formed by the three points is the triangle of the vertex of the inner boundary.
④ when the included angle of the vertex of an inner boundary is greater than or equal to the simplified angle A, the included angle of the vertex is set V from the point2And VsIs removed. When the included angle of the vertex of a certain outer boundary is larger than or equal to the simplified angle A, the vertex of the included angle of the vertex is selected from the point set V1And VsIs removed. When the ratio of the area of the outer boundary vertex triangle to the area of the outer boundary enclosure is less than or equal to the reducible area ratio, one vertex of the outer boundary vertex triangle is selected from the point set V1And VsRemoving the vertex in the point set VsOr V1Between the other two vertices. When the ratio of the triangle area of the inner boundary vertex to the area enclosed by the inner boundary is less than or equal to the area ratio which can be simplified, one vertex of the triangle of the inner boundary vertex is collected from the points V2And VsRemoving the vertex in the point set VsOr V2Between the other two vertices.
Fifthly, repeating the third step and the fourth step until the fourth step does not remove any point.
⑥ simplified point set V after ⑤sThe connecting line between every two adjacent points in the image is a simplified edge set Es。
b. Rejoining the oversimplification point:
(1) the simplified plane graph is put into a coordinate system oxyz. Point viHas a coordinate value of (x)vi,yvi) U, point uiHas a coordinate value of (x)ui,yui)。
(2) Simplified edge set EsTwo end points v of one edges,vqAnd q > s. If q-s > 1, then at point vsAnd point vqWith q-s-1 points in between from the set of points V of step ①sIs removed. If point set Vi＝{vs,vs+1,...,vqThe abscissa and ordinate of the point set are not increased and not decreased, then the point set V is obtainediRe-adding the point set V after ⑤s。
(3) Point set V after (2)sThe connecting lines between two adjacent points in the edge list form a new simplified edge set Es。
(4) Simplified edge set E after (3)sTwo end points v of one edges,vqAnd q > s. If q-s > 1, then at point vsAnd point vqWith q-s-1 points in between from the set of points V of step ①sIs removed. Taking point vrAnd s < r < q. If point vrPoint vsAnd point vqFormed by point vrThe angle of the apex is less than 90 deg., point vrRe-adding the point set V after ⑤s。
(5) Point set V after (2) - (4)sThe connecting lines between two adjacent points in the edge list form a new simplified edge set Es. Let Ms＝(Vs,Es)。
Step 4, simplifying edge set E after step 5sTwo end points v of one edges,vqAnd q > s. If q-s > 1, the point removed is made straight perpendicular to the point vsPoint vqConnecting the straight line, connecting the straight line with the point vsPoint vqReplacing points corresponding to the point set V by the pendants of the connected straight lines to obtain a subdomain decomposition input boundary Md. Decomposing input boundaries M for sub-fieldsdA subfield decomposition is performed to obtain a subfield boundary M'.
And 5, generating topology, which comprises the following specific steps:
and 5.1 determining corner points.
Real top points and virtual top points exist in the subdomain boundary M', and the real top points are the simplified edge set E after (5)sThe virtual vertex is a point other than the real vertex.
The sub-domain boundaries having more than 4 real vertices need to be simplified again for each sub-domain boundary composed of only the real vertices, the simplification method is that the reducible angle a is set to 90 degrees, the reducible area ratio γ is set to 0.1, and when the included angle of a vertex of a certain inner boundary is greater than or equal to the reducible angle a, the vertex of the included angle is removed from the sub-domain boundaries. When the included angle of the vertex of a certain outer boundary is larger than or equal to the simplified angle A, the vertex of the included angle of the vertex is removed from the boundary of the subdomain. When the ratio of the area of the outer-boundary vertex triangle to the area of the subdomain boundary is less than or equal to the reducible area ratio, one vertex of the outer-boundary vertex triangle is removed from the subdomain boundary, and the removed vertex is located between the other two vertices in the subdomain boundary. When the ratio of the triangle area of the inner boundary vertex to the boundary area of the sub-region is less than or equal to the reducible area ratio, one vertex of the triangle of the inner boundary vertex is removed from the boundary of the sub-region, and the removed vertex is located between the other two vertices in the boundary of the sub-region. The remaining real vertices are the corner points.
5.2 determining which topological mode is used to generate the topological structure of the given boundary region by calculating the number of edges of each edge of the boundary of the subdomain in the step 4; the topological modes include a rectangular area topological mode and a triangular area topological mode.
And 6, realizing grid fairing by adopting a Laplace fairing algorithm.
6.1 the coordinates of the four vertices of the rectangular region are (0,0), (1,1), and (0,1), and vertex position coordinate information is determined for points on the boundary of the parameter domain of the rectangular region based on the weights. The weight calculation method is as follows: let the vertex sequence of a certain edge side of the parameter domain be w0,w1,...,wt-1The number of vertices at side is t, and the vertex sequence at side in the corresponding subfield boundary is w'0,w′1,...,w′t-1. Connecting two vertices w'0And w't-1Line segment s, passing point w'1,w′2,...,w′t-2Making a perpendicular line to the line segment s, and intersecting the line segment s with the point w ″1,w″2,...,w″t-2Then the weight wgt of each vertex in the side of the parameter field is equal to the corresponding vertex of the vertex to w 'in the subdomain boundary'0Distance from vertex w't-1To vertex w'0The distance of (c).
6.2 supposing the coordinates of three corner points of the triangular region are (0,0), (1,0), (0.5,1), adding the upper side to the triangular region, and making the corner point with the coordinate position of (0.5,1) be C1，C1The corresponding vertex in the subdomain boundary is C'1(ii) a Adding one vertex C'2Make vertex C'1And vertex C'2Having the same coordinate position. Then vertex C'2Addition to the parameter field corresponds to C2Let vertex C1And vertex C2Has the coordinates of (1,0) and (0,1), respectively, and thus contains a vertex C in the parameter domain1The four-sided mesh of (2) would become a five-sided mesh, but with vertex C1And vertex C2Both correspond to the same coordinate position in the subdomain boundary, and the two vertices are mapped to the same vertex in the subdomain boundary during mapping, so the mesh still maintains a four-sided mesh after mapping. And determining vertex position coordinate information of points on the boundary of the triangular region according to the weight, wherein the weight calculation method is the same as that of the rectangular region.
6.3 the vertex position coordinate information on the parameter boundary is determined as follows:
the vertex coordinates on the right boundary are represented by w ═ 1.0, wgtw) The vertex coordinates on the lower boundary are represented by w ═ wgtw0.0), and the vertex coordinates on the left-hand boundary are represented by w ═ 0.0, wgtw) The vertex coordinates on the upper boundary are represented by w ═ w (wgt)w,1.0)。
6.4 smoothing the internally generated topology vertexes based on the laplacian smoothing algorithm to obtain the coordinate information of each internal vertex, which is specifically as follows:
for any one of the internal vertex coordinates O, the smoothed position coordinate information is as follows:
Figure BDA0001293647390000051
wherein, t1Indicating the number of vertices directly connected to the internal vertex, OjRepresenting the coordinates of the vertex directly connected to the internal vertex. And listing the position coordinate information equation after fairing for all internal vertexes, thereby constructing a linear equation set, and solving to obtain the coordinate information of each vertex in the parameter domain grid.
And 7, mapping the generated four-side grids in the parameter domain to the subdomain boundary after the step 5.1, wherein the mapping method is an overrun interpolation mapping method. And the triangular area maps two vertexes on the upper side into one vertex during mapping.
The invention has the following beneficial effects:
the invention provides a high-quality quadrilateral mesh subdivision method. The quadrilateral mesh generated by the method has fewer singular points, has good smoothness, uniformity and orthogonality, and can be directly applied to finite element analysis simulation solution.
Drawings
FIG. 1 is a schematic view of a cross-sectional apex distribution taken after geometric dissection;
FIG. 2 is a schematic diagram of vertex distribution for transforming a multi-connected domain into a single-connected domain;
FIG. 3 is a simplified distribution diagram of vertices;
FIG. 4 is a schematic diagram of the vertex distribution after adding the overcomplification point;
FIG. 5 is a schematic diagram of the vertex distribution after dividing the subdomain boundaries;
FIG. 6 is a schematic diagram of a topology of rectangular areas;
FIG. 7 is a schematic diagram of the topology structure in FIG. 6 after mesh fairing is implemented by using a Laplace fairing algorithm;
FIG. 8 is a schematic view of a topology of triangular regions;
FIG. 9 is a schematic diagram of the topology after the triangular region in FIG. 8 is transformed into a rectangular region;
FIG. 10 is a quadrilateral segmentation map after a quadrilateral mesh is mapped to a subdomain boundary.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
The quadrilateral segmentation method applied to finite element analysis comprises the following specific steps:
step 1, as shown in fig. 1, a geometric model is introduced into finite element analysis software, a geometric body with a through hole is cut open, and a cross section obtained after the geometric body is cut open is a plane figure which is provided with an inner boundary and an outer boundary, wherein the inner boundary and the outer boundary are both polygons or are formed by adopting polygon fitting.
Step 2, as shown in FIG. 2, converting the multi-connected domain into a single-connected domain;
the multiply-connected domain described in the present invention is applicable to only one inner boundary.
The set of all the vertexes on the inner boundary of the obtained plane graph is a point set V1The set of all the top points on the outer boundary is the point set V2. Set of points V1And V2Find point v thereina、ubL (v) of the othera,ub) Is the smallest of all inner-boundary vertex-to-outer-boundary vertex distances, i.e.:
l(va,ub)＝min(l(vi,uj)),vi∈V1,uj∈V2
wherein i is more than or equal to 0 and less than or equal to n1-1, j is more than or equal to 0 and less than or equal to n2-1, n1 is an integer more than or equal to 3, and n2 is an integer more than or equal to n 1.
In obtaining va,ubThen, from the set of points:
V1＝{v0,v1,...,vn1-1}
V2＝{u0,u1,...,un2-1}
the sequences of the point set V' are obtained:
V′＝{v0,v1,...,va,ub,ub+1,...,un2-1,u0,u1,...,ub,va,va+1,...,vn1-1}
then the edge (v)a,ub) And (u)b,va) Is a pair of newly added edges with overlapped positions, and the number of the points in the point set V' is as follows: n is n1+n2+2. Let the set of all edges of the plane graph be the edge set E. Let simply connect the boundary M ═ V', E.
In order to keep the density between the vertexes in the point set V' uniform, the opposite side (V)a,ub) And side (u)b,va) Performing uniform interpolation, i.e. at the edge (v)a,ub) And side (u)b,va) A new set of vertices Z is added. l (v)a,ub) The value m divided by the average length of all edges in the edge set E is calculated as follows:
Figure BDA0001293647390000061
where p is (i +1) mod (n1), q is (i +1) mod (n2), mod represents a remainder calculation, and the edge set E is a set of edges1Is V1Set of lengths of the lines between adjacent points in (E)2Is V2The length of the line segment between each adjacent point in the set, the number m of the top points of the top point set Z1Rounding off the rounding value for m.
Sequence of point set V:
Figure BDA0001293647390000062
in the formula (I), the compound is shown in the specification,
Figure BDA0001293647390000063
are vertices in the set of vertices Z.
And 3, simplifying the plane graph, which comprises the following specific steps:
step a, as shown in fig. 3, deleting the simplifiable vertex:
① order simplified point set Vs＝V。
② two thresholds for controlling the degree of simplification of the boundary are defined as a simplified angle A and a simplified area ratio gamma, wherein A is more than or equal to 0 and less than or equal to pi, and gamma is more than or equal to 0 and less than or equal to 0.25.
③ set of points V1At any point viAnd two adjacent vertexes with viIs the apex angle αiThe included angle of the vertex of the outer boundary is formed, and a triangle formed by the three points is the triangle of the vertex of the outer boundary; set of points V2At any point ujAnd two adjacent vertices by point ujIs the apex angle βjThe included angle of the vertex of the inner boundary is shown, and the triangle formed by the three points is the triangle of the vertex of the inner boundary.
④ when the included angle of the vertex of an inner boundary is greater than or equal to the simplified angle A, the included angle of the vertex is set V from the point2And VsIs removed. When the included angle of the vertex of a certain outer boundary is larger than or equal to the simplified angle A, the vertex of the included angle of the vertex is selected from the point set V1And VsIs removed. When the ratio of the area of the outer boundary vertex triangle to the area of the outer boundary enclosure is less than or equal to the reducible area ratio, one vertex of the outer boundary vertex triangle is selected from the point set V1And VsRemoving the vertex in the point set VsOr V1Between the other two vertices. When the ratio of the triangle area of the inner boundary vertex to the area enclosed by the inner boundary is less than or equal to the area ratio which can be simplified, one vertex of the triangle of the inner boundary vertex is collected from the points V2And VsRemoving the vertex in the point set VsOr V2Between the other two vertices.
Fifthly, repeating the third step and the fourth step until the fourth step does not remove any point.
⑥ simplified point set V after ⑤sThe connecting line between every two adjacent points in the image is a simplified edge set Es。
Step b, as shown in fig. 4, adding the over-simplification point again:
(1) the simplified plane graph is put into a coordinate system oxyz. Point viHas a coordinate value of (x)vi,yvi) U, point uiHas a coordinate value of (x)ui,yui)。
(2) Simplified edge set EsTwo end points v of one edges,vqAnd q > s. If q-s > 1, then at point vsAnd point vqWith q-s-1 points in between from the set of points V of step ①sIs removed. If point set Vi＝{vs,vs+1,...,vqThe abscissa and ordinate of the point set are not increased and not decreased, then the point set V is obtainediRe-adding the point set V after ⑤s。
(3) Point set V after (2)sThe connecting lines between two adjacent points in the edge list form a new simplified edge set Es。
(4) Simplified edge set E after (3)sTwo end points v of one edges,vqAnd q > s. If q-s > 1, then at point vsAnd point vqWith q-s-1 points in between from the set of points V of step ①sIs removed. Taking point vrAnd s < r < q. If point vrPoint vsAnd point vqFormed by point vrThe angle of the apex is less than 90 deg., point vrRe-adding the point set V after ⑤s。
(5) Point set V after (2) - (4)sThe connecting lines between two adjacent points in the edge list form a new simplified edge set Es. Let Ms＝(Vs,Es)。
Step 4, simplifying edge set E after step 5sTwo end points v of one edges,vqAnd q > s. If q-s > 1, the point removed is made straight perpendicular to the point vsPoint vqConnecting the straight line, connecting the straight line with the point vsPoint vqReplacing points corresponding to the point set V by the pendants of the connected straight lines to obtain a subdomain decomposition input boundary Md. Decomposing input boundaries M for sub-fieldsdPerforming subfield decomposition to obtain a subfield boundary M', as shown in fig. 5; the specific calculation method of the subdomain decomposition adopts a simple polygon convex unit subdivision algorithm, which is reported by Yanshan university, Gaixiang, 2004 and 7 months.
Reducing the geometric feature points of the boundary is also the main purpose of the original boundary decomposition in this section. The geometric shape of the boundary is relatively simple, the complexity of subsequent steps of generating a topological structure, grid mapping and the like is reduced, and the quality of the topological grid can be improved.
Step 5, as shown in fig. 6, 7, 8 and 9, generating a topology, specifically including the steps of:
and 5.1 determining corner points.
Real top points and virtual top points exist in the subdomain boundary M', and the real top points are the simplified edge set E after (5)sThe virtual vertex is a point other than the real vertex. To be able to determine the corner points, the number of real vertices of these subdomain boundaries is guaranteed to be 3 or 4. A subdomain boundary has only 3 or 4 real vertices, which are corner points.
The sub-domain boundaries having more than 4 real vertices need to be simplified again for each sub-domain boundary composed of only the real vertices, the simplification method is that the reducible angle a is set to 90 degrees, the reducible area ratio γ is set to 0.1, and when the included angle of a vertex of a certain inner boundary is greater than or equal to the reducible angle a, the vertex of the included angle is removed from the sub-domain boundaries. When the included angle of the vertex of a certain outer boundary is larger than or equal to the simplified angle A, the vertex of the included angle of the vertex is removed from the boundary of the subdomain. When the ratio of the area of the outer-boundary vertex triangle to the area of the subdomain boundary is less than or equal to the reducible area ratio, one vertex of the outer-boundary vertex triangle is removed from the subdomain boundary, and the removed vertex is located between the other two vertices in the subdomain boundary. When the ratio of the triangle area of the inner boundary vertex to the boundary area of the sub-region is less than or equal to the reducible area ratio, one vertex of the triangle of the inner boundary vertex is removed from the boundary of the sub-region, and the removed vertex is located between the other two vertices in the boundary of the sub-region. The remaining real vertices are the corner points.
5.2 determining which topological mode is used to generate the topological structure of the given boundary region by calculating the number of edges of each edge of the boundary of the subdomain in the step 4; the topological pattern is as follows: 4 topological modes suitable for rectangular areas and 3 topological modes suitable for triangular areas; the determination of which topological mode to use and the specific generation method of the topological structure under each topological mode adopt the processing method in "Pattern-Based scheduling for N-sized Patches, Kenshi Takayama Daniele Panozzo Olga Sorkine-horning, Europatics Symposium on GeometryProcessing 2014".
And 6, realizing grid fairing by adopting a Laplace fairing algorithm.
6.1 assume that the coordinates of the four vertices of the rectangular area are (0,0), (1,1), and (0,1), i.e. the rectangular area is a square of one unit, which is convenient for the mapping calculation in the subsequent steps. In order to accomplish mapping on the boundary as well, vertex position coordinate information is determined for points on the rectangular region parameter domain boundary according to the weights. The weight calculation method is as follows: let the vertex sequence of a certain edge side of the parameter domain be w0,w1,...,wt-1The number of vertices at side is t, and the vertex sequence at side in the corresponding subfield boundary is w'0,w′1,...,w′t-1. Connecting two vertices w'0And w't-1Line segment s, passing point w'1,w′2,...,w′t-2Making a perpendicular line to the line segment s, and intersecting the line segment s with the point w ″1,w″2,...,w″t-2Then the weight wgt of each vertex in the side of the parameter field is equal to the corresponding vertex of the vertex to w 'in the subdomain boundary'0Distance from vertex w't-1To vertex w'0From the above results, it is found that 0. ltoreq. wgt. ltoreq.1.
6.2 supposing that the coordinates of three corner points of the triangular area are (0,0), (1,0), (0.5,1), the triangular area has less upper sides than the rectangular area, so that the upper sides are added to the triangular area, and the corner point with the coordinate position of (0.5,1) is C1，C1The corresponding vertex in the subdomain boundary is C'1(ii) a Adding one vertex C'2Make vertex C'1And vertex C'2Having the same coordinate position. Then vertex C'2Addition to the parameter field corresponds to C2Let vertex C1And vertex C2Has the coordinates of (1,0) and (0,1), respectively, and thus contains a vertex C in the parameter domain1The four-sided mesh of (2) would become a five-sided mesh, but with vertex C1And vertex C2All corresponding to the sub-field boundariesAnd the two vertexes are mapped to the same vertex in the subdomain boundary during mapping, so that the mesh still always keeps a quadrilateral mesh after mapping. The triangular region in fig. 8 is transformed into a rectangular region as shown in fig. 9. And determining vertex position coordinate information of points on the boundary of the triangular region according to the weight, wherein the weight calculation method is the same as that of the rectangular region.
6.3 the vertex position coordinate information on the parameter boundary is determined as follows:
the vertex coordinates on the right boundary are represented by w ═ 1.0, wgtw) The vertex coordinates on the lower boundary are represented by w ═ wgtw0.0), and the vertex coordinates on the left-hand boundary are represented by w ═ 0.0, wgtw) The vertex coordinates on the upper boundary are represented by w ═ w (wgt)w,1.0). Taking the divided rectangular area in fig. 6, the determined vertex coordinates are as shown in fig. 7.
6.4 smoothing the internally generated topology vertexes based on the laplacian smoothing algorithm to obtain the coordinate information of each internal vertex, which is specifically as follows:
for any one of the internal vertex coordinates O, the smoothed position coordinate information is as follows:
Figure BDA0001293647390000091
wherein, t1Indicating the number of vertices directly connected to the internal vertex, OjRepresenting the coordinates of the vertex directly connected to the internal vertex. And listing the position coordinate information equation after fairing for all internal vertexes, thereby constructing a linear equation set, and solving to obtain the coordinate information of each vertex in the parameter domain grid.
And 7, mapping the generated four-side grids in the parameter domain to the subdomain boundary after the step 5.1, wherein the mapping method is an overrun interpolation mapping method. The triangular area maps two vertexes on the upper side into one vertex in mapping. The mapped quad mesh is shown in fig. 10.

Claims (1)

1. The quadrilateral segmentation method applied to finite element analysis is characterized by comprising the following steps of: the method comprises the following specific steps:
step 1, introducing a geometric model into finite element analysis software, splitting a geometric body with a through hole, wherein a cross section obtained after the geometric body is split is a plane figure which has only one inner boundary and one outer boundary, and the inner boundary and the outer boundary are both polygons or are formed by adopting polygon fitting;
step 2, converting the multi-connected domain into a single-connected domain;
the set of all the vertexes on the inner boundary of the obtained plane graph is a point set V1The set of all the top points on the outer boundary is the point set V2(ii) a Set of points V1And V2Find point v thereina、ubL (v) of the othera,ub) Is the smallest of all inner-boundary vertex-to-outer-boundary vertex distances, i.e.:
l(va,ub)＝min(l(vi,uj)),vi∈V1,uj∈V2
wherein i is more than or equal to 0 and less than or equal to n1-1, j is more than or equal to 0 and less than or equal to n2-1, n1 is an integer more than or equal to 3, and n2 is an integer more than or equal to n 1;
in obtaining va,ubThen, from the set of points:
V1＝{v0,v1,...,vn1-1}
V2＝{u0,u1,...,un2-1}
the sequences of the point set V' are obtained:
V′＝{v0,v1,...,va,ub,ub+1,...,un2-1,u0,u1,...,ub,va,va+1,...,vn1-1}
then the edge (v)a,ub) And (u)b,va) Is a pair of newly added edges with overlapped positions, and the number of the points in the point set V' is as follows: n-n 1+ n2+ 2; making the set of all edges of the plane graph as an edge set E; let the single connectivity boundary M ═ (V', E);
opposite side (v)a,ub) And side (u)b,va) Performing uniform interpolation, i.e. at the edge (v)a,ub) And side (u)b,va) Adding a new vertex set Z; l (v)a,ub) The value m divided by the average length of all edges in the edge set E is calculated as follows:
Figure FDA0002594624050000011
where p is (i +1) mod (n1), q is (i +1) mod (n2), mod represents a remainder calculation, and the edge set E is a set of edges1Is V1Set of lengths of the lines between adjacent points in (E)2Is V2The length of the line segment between each adjacent point in the set, the number m of the top points of the top point set Z1Rounding off the rounding value for m;
sequence of point set V:
Figure FDA0002594624050000012
in the formula (I), the compound is shown in the specification,
Figure FDA0002594624050000021
the vertexes in the vertex set Z;
and 3, simplifying the plane graph, which comprises the following specific steps:
a. deleting may simplify vertices:
① order simplified point set Vs＝V；
Secondly, defining two thresholds for controlling the simplification degree of the boundary as a simplified angle A and a simplified area ratio gamma, wherein A is more than or equal to 0 and less than or equal to pi, and gamma is more than or equal to 0 and less than or equal to 0.25;
③ set of points V1At any point viAnd two adjacent vertexes with viIs the apex angle αiThe included angle of the vertex of the outer boundary is formed, and a triangle formed by the three points is the triangle of the vertex of the outer boundary; set of points V2At any point ujAnd two adjacent vertices by point ujIs the apex angle βjIs the included angle of the inner boundary vertexThe triangle formed by the points is an inner boundary vertex triangle;
④ when the included angle of the inner boundary vertex is greater than or equal to the simplified angle A, the included angle of the inner boundary vertex is from the point set V2And VsRemoving; when the included angle of the vertex of a certain outer boundary is larger than or equal to the simplified angle A, the included angle of the vertex of the outer boundary is determined from the point set V1And VsRemoving; when the ratio of the area of the outer boundary vertex triangle to the area of the outer boundary enclosure is less than or equal to the reducible area ratio, one vertex of the outer boundary vertex triangle is selected from the point set V1And VsRemoving the vertex in the point set VsOr V1Between the other two vertices; when the ratio of the triangle area of the inner boundary vertex to the area enclosed by the inner boundary is less than or equal to the area ratio which can be simplified, one vertex of the triangle of the inner boundary vertex is collected from the points V2And VsRemoving the vertex in the point set VsOr V2Between the other two vertices;
fifthly, repeating the third step and the fourth step until the fourth step is finished without removing any points;
⑥ simplified point set V after ⑤sThe connecting line between every two adjacent points in the image is a simplified edge set Es；
b. Rejoining the oversimplification point:
(1) putting the simplified plane graph into a coordinate system oxyz; point viHas a coordinate value of (x)vi,yvi) U, point uiHas a coordinate value of (x)ui,yui)；
(2) Simplified edge set EsTwo end points v of one edges,vqQ > s; if q-s > 1, then at point vsAnd point vqWith q-s-1 points in between from the set of points V of step ①sRemoving; if point set Vi＝{vs,vs+1,...,vqThe abscissa and ordinate of the point set are not increased and not decreased, then the point set V is obtainediRe-adding the point set V after ⑤s；
(3) Point set V after (2)sThe connecting lines between two adjacent points in the system are recombined into a new simplified edge setEs；
(4) Simplified edge set E after (3)sTwo end points v of one edges,vqQ > s; if q-s > 1, then at point vsAnd point vqWith q-s-1 points in between from the set of points V of step ①sRemoving; taking point vrR is more than s and less than q; if point vrPoint vsAnd point vqFormed by point vrThe angle of the apex is less than 90 deg., point vrRe-adding the point set V after ⑤s；
(5) Point set V after (2) - (4)sThe connecting lines between two adjacent points in the edge list form a new simplified edge set Es(ii) a Let Ms＝(Vs,Es)；
Step 4, simplifying edge set E after step 5sTwo end points v of one edges,vqQ > s; if q-s > 1, the point removed is made straight perpendicular to the point vsPoint vqConnecting the straight line, connecting the straight line with the point vsPoint vqReplacing points corresponding to the point set V by the pendants of the connected straight lines to obtain a subdomain decomposition input boundary Md(ii) a Decomposing input boundaries M for sub-fieldsdPerforming sub-domain decomposition to obtain a sub-domain boundary M';
and 5, generating topology, which comprises the following specific steps:
5.1 determining a corner point;
real top points and virtual top points exist in the subdomain boundary M', and the real top points are the simplified edge set E after (5)sA virtual vertex is a point other than the real vertex;
the sub-domain boundaries with more than 4 real vertexes need to be simplified again, the simplified method is that the simplified angle A is set to be 90 degrees, the simplified area ratio gamma is set to be 0.1, and when the included angle of the vertexes of a certain inner boundary is larger than or equal to the simplified angle A, the vertexes of the included angle of the vertexes of the inner boundary are removed from the sub-domain boundaries; when the included angle of the vertex of one outer boundary is larger than or equal to the simplified angle A, removing the vertex of the included angle of the vertex of the outer boundary from the subdomain boundary; when the ratio of the area of the outer boundary vertex triangle to the area of the subdomain boundary is less than or equal to the reducible area ratio, removing one vertex of the outer boundary vertex triangle from the subdomain boundary, wherein the removed vertex is positioned between the other two vertices in the subdomain boundary; when the ratio of the triangle area of the inner boundary vertex to the boundary area of the sub-domain is less than or equal to the area ratio, removing one vertex of the triangle of the inner boundary vertex from the boundary of the sub-domain, wherein the removed vertex is positioned between the other two vertices in the boundary of the sub-domain; the remaining real vertices are angular points;
5.2 determining which topological mode is used to generate the topological structure of the given boundary region by calculating the number of edges of each edge of the boundary of the subdomain in the step 4; the topological modes comprise a rectangular area topological mode and a triangular area topological mode;
step 6, realizing grid fairing by adopting a Laplace fairing algorithm;
6.1, assuming that the coordinates of four vertexes of the rectangular area are (0,0), (1,1) and (0,1), determining vertex position coordinate information according to the weight for points on the parameter domain boundary of the rectangular area; the weight calculation method is as follows: let the vertex sequence of a certain edge side of the parameter domain be w0,w1,...,wt-1The number of vertices at side is t, and the vertex sequence at side in the corresponding subfield boundary is w'0,w′1,...,w′t-1(ii) a Connecting two vertices w'0And w't-1Line segment s, passing point w'1,w′2,...,w′t-2Making a perpendicular line to the line segment s, and intersecting the line segment s with the point w ″1,w″2,...,w″t-2Then the weight wgt of each vertex in the side of the parameter field is equal to the corresponding vertex of the vertex to w 'in the subdomain boundary'0Distance from vertex w't-1To vertex w'0The ratio of the distances of (a);
6.2 supposing the coordinates of three corner points of the triangular region are (0,0), (1,0), (0.5,1), adding the upper side to the triangular region, and making the corner point with the coordinate position of (0.5,1) be C1，C1The corresponding vertex in the subdomain boundary is C'1(ii) a Adding one vertex C'2Make vertex C'1And vertex C'2Have the same seatMarking a position; then vertex C'2Addition to the parameter field corresponds to C2Let vertex C1And vertex C2Has the coordinates of (1,0) and (0,1), respectively, and thus contains a vertex C in the parameter domain1The four-sided mesh of (2) would become a five-sided mesh, but with vertex C1And vertex C2The two vertexes are mapped to the same vertex in the subdomain boundary during mapping, so that the grid still keeps a four-side grid after mapping; determining vertex position coordinate information of points on the boundary of the triangular region according to the weight, wherein the weight calculation method is the same as that of the rectangular region;
6.3 the vertex position coordinate information on the parameter boundary is determined as follows:
the vertex coordinate on the right-hand boundary is denoted by w1The vertex coordinate on the lower side boundary is represented as w (1.0, wgt)2The vertex coordinate on the left boundary is represented as w ═ wgt,0.03(0.0, wgt), and the vertex coordinates on the upper boundary are represented as w4＝(wgt,1.0)；
6.4 smoothing the internally generated topology vertexes based on the laplacian smoothing algorithm to obtain the coordinate information of each internal vertex, which is specifically as follows:
for any one of the internal vertex coordinates O, the smoothed position coordinate information is as follows:
Figure FDA0002594624050000041
wherein, t1Indicating the number of vertices directly connected to the internal vertex, OjCoordinates representing vertices directly connected to the internal vertex; listing the position coordinate information equations after fairing for all internal vertexes, thereby constructing a linear equation set, and solving to obtain the coordinate information of each vertex inside the parameter domain grid;
step 7, mapping the generated quadrilateral grids in the parameter domain to the subdomain boundary after the step 5.1, wherein the mapping method is an overrun interpolation mapping method; and the triangular area maps two vertexes on the upper side into one vertex during mapping.
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